How can the integrity of a system backup be determined by an ordinary user such as myself? Or is it even possible?
Two words.
Test Restore.
(1) Take Backup Disk
(2) Pick "random" selection of files located on backup disk
(3) Restore said files from backup
(4) Look at files with Mk1 Eyeball
(5) Come to your own conclusions
(6) Rinse and repeat where N = Qty Backup Disks
NEVER use automated processes to check backup integrity. There is no substitute for the human eyeball in fighting stuff like silent corruption or bitrot.
If you want to be really serious about it , you would also store cryptographic hashes of your files on separate storage devices (or even printed on paper). You would then compare the stored hash against the hash of the restored file. This would then give you a guarantee of original and backup being identical, and would also assist when comparing binary files (such as images), where its difficult to review by eye.